Complete intersections of quadrics and complete intersections on Segre varieties with common specializations

Abstract

. We investigate whether surfaces that are complete intersections of quadrics and complete intersection surfaces in the Segre embedded product P 1 × P k ֒ → P 2 k +1 can belong to the same Hilbert scheme. For k = 2 there is a classical example; it comes from K3 surfaces in projective 5-space that degenerate into a hypersurface on the Segre threefold. We show that for k ≥ 3 there is only one more example. It turns out that its (connected) Hilbert scheme has at least two irreducible components. We investigate the corresponding local moduli